There is a HAPpy Banach space, other than l_2, which has a symmetric basis

You need to know: Hilbert space $l_2$ , Banach space B, notation $\|.\|_B$ for norm in B, convergence in this norm, isomorphic Banach spaces, dimension of a Banach space, finite and infinite dimensional Banach spaces, subspace of a Banach space, bounded linear operator K between Banach spaces, range of K, compact set in a Banach space.

Background: A sequence $\{x_n\}$ of a Banach space B is called basis of B if every $x\in B$ has a unique representation of the form $x=\sum\limits_{n=1}^\infty a_nx_n$ for some real numbers $a_n$ . Two bases $\{x_n\}$ and $\{y_n\}$ of B are called equivalent if series $x=\sum\limits_{n=1}^\infty a_nx_n$ converges in B if and only if $x=\sum\limits_{n=1}^\infty a_ny_n$ converges. A basis $\{x_n\}$ of B is called symmetric if every permutation of $\{x_n\}$ is a basis of B equivalent to $\{x_n\}$ .

A Banach space B is said to have the approximation property (AP) if for every compact set K in B and for every $\epsilon>0$ , there is a bounded linear operator $T:B \to B$ , whose range is finite-dimensional, and such that $\|Tx-x\|_B \leq \epsilon$ for all $x \in K$ . We say that Banach space B has the hereditary AP (or is a HAPpy space) if all of its subspaces have the AP.

The Theorem: On 7th November 2010, William Johnson and Andrzej Szankowski submitted to the Annals of Mathematics a paper in which they proved the existence of a HAPpy Banach space, not isomorphic to the Hilbert space $l_2$ , which has a symmetric basis.

Short context: The first examples of HAPpy Banach spaces not isomorphic to a Hilbert space was constructed by Johnson in 1980. Later, Pisier constructed more such examples, but none of them have a symmetric basis. In fact, Johnson asked in 1980 whether there exists any HAPpy space, other than $l_2$ , that has a symmetric basis. The Theorem answers this question affirmatively.

Links: The original paper is available here.

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The size of subset of {1,…,N} without 3-term arithmetic progressions is O(N/(log n)^(1-o(1)))

You need to know: A (nontrivial) 3-term arithmetic progression, big O notation, small o notation.

Background: For integer $N>0$ , let $r_3(N)$ be the cardinality of the largest subset of $\{1, 2, \dots , N\}$ which contains no nontrivial 3-term arithmetic progressions.

The Theorem: On 30th October 2010, Tom Sanders submitted to arxiv a paper in which he proved that $r_3(N) = O\left(\frac{N(\log\log N)^6}{\log N}\right)$ .

Short context: In 1936, Erdős and Turán conjectured that any set containing a positive proportion of integers must contain a 3-term arithmetic progression. This is equivalent to $\lim\limits_{N\to\infty}\frac{r_3(N)}{N}=0$ . In 1953, Roth confirmed this conjecture by proving that $r_3(N) = O\left(\frac{N}{\log\log N}\right)$ . In 1999, Bourgain improved this to $r_3(N) = O\left(\frac{N\sqrt{\log\log N}}{\sqrt{\log N}}\right)$ . The Theorem proves a much stronger upper bound, the first one in the form $r_3(N) = O\left(\frac{N}{\log^{1-o(1)} N}\right)$ . In a later work, Bloom improved the bound slightly to $r_3(N) = O\left(\frac{N(\log\log N)^4}{\log N}\right)$ .

Links: Free arxiv version of the original paper is here, journal version is here.

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Internal DLA process converges to a disk with at most logarithmic discrepancy

You need to know: Integer lattice ${\mathbb Z}^2$ , origin $0$ , basic probability theory, simple random walk on ${\mathbb Z}^2$ , notation $\left \lfloor{t}\right \rfloor$ be the largest integer not exceeding t.

Background: The internal diffusion limited aggregation process (internal DLA process for short) is defined as follows. For each integer time $n\geq 1$ , construct a random set $A(n) \subset {\mathbb Z}^2$ such that (i) $A(1) = \{0\}$ , and (ii) for each $n\geq 1$ , let $A(n+1)$ be $A(n)$ plus the first point at which a simple random walk from the origin hits ${\mathbb Z}^2 \setminus A(n)$ . For any real $t\geq 1$ , let $A(t)=A(\left \lfloor{t}\right \rfloor)$ . For $r>0$ , let $B(r)=\{z=(z_1,z_2) \in {\mathbb Z}^2\,|\,z_1^2+z_2^2 < r^2\}$ .

The Theorem: On 12th October 2010, David Jerison, Lionel Levine, and Scott Sheffield submitted to arxiv a paper in which they proved the existence of an absolute constant C such that with probability 1, $B(r-C\log r) \subset A(\pi r^2) \subset B(r+C\log r)$ for all sufficiently large r.

Short context: Internal DLA process was proposed by Meakin and Deutch in 1986 as a model of industrial chemical processes. They found numerically that, for large n, the process becomes close to a disk with at most logarithmic fluctuations. In 1992, Lawler, Bramson and Griffeath proved that the asymptotic shape of the domain $A(n)$ is indeed a disk. The Theorem confirms that the fluctuations are indeed at most logarithmic.

Links: Free arxiv version of the paper is here, journal version is here.

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The distortion of a knot in R^3 can be arbitrary large

You need to know: Euclidean space ${\mathbb R}^3$ , Euclidean distance in ${\mathbb R}^3$ , curve in ${\mathbb R}^3$ , closed curves, self-intersecting and non-self-intersecting curves, arclength along the curve.

Background: A knot representative is a closed, non-self-intersecting curve $\gamma$ in ${\mathbb R}^3$ . Two knot representatives $\gamma_1$ and $\gamma_2$ are equivalent if $\gamma_1$ can be transformed smoothly, without intersecting itself, to coincide with $\gamma_2$ . A knot K is the set of all knot representatives $\gamma$ equivalent to a fixed knot representative $\gamma_0$ . The distortion of knot representative $\gamma$ is $\delta(\gamma)=\sup\limits_{p,q \in \gamma}\frac{d_\gamma(p,g)}{d_{{\mathbb R}^3}(p,q)}$ , where $d_\gamma$ denotes the arclength along $\gamma$ and $d_{{\mathbb R}^3}$ denotes the Euclidean distance in ${\mathbb R}^3$ . The distortion of a knot K is $\delta(K) = \inf\limits_{\gamma \in K} \delta(\gamma)$ .

The Theorem: On 10th October 2010, John Pardon submitted to arxiv a paper in which he proved that for any $C>0$ there exists a knot K in ${\mathbb R}^3$ with $\delta(K) > C$ .

Short context: The question whether every knot has a “nice” representative is well-studied for various definitions of “nice”, see here for an example. In 1983, Gromov asked whether every knot K in ${\mathbb R}^3$ have a representative $\gamma$ with $\delta(\gamma) < 100$ . The Theorem shows that this is not the case, with any constant in place of $100$ . In fact, Pardon presented explicit examples of families of knots with distortion going to infinity. For example, let $\gamma_{p,q}$ be the curve in ${\mathbb R}^3$ defined by equations $x=r\cos(p\phi)$ , $y=r\sin(p\phi)$ , $z=-\sin(q\phi)$ , where $r=\cos(q\phi)+2$ and $0\leq \phi <2\pi$ . Set $K_{p,q}$ of curves equivalent to $\gamma_{p,q}$ is called the $(p,q)$ -torus knot. Pardon proved that $\delta(K_{p,q})\geq \frac{1}{160}\min(p,q)$ . The Theorem follows.

Links: Free arxiv version of the original paper is here, journal version is here.

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For each N>c_d t^d, there exists an N-point spherical t-design in the sphere S^d

You need to know: Notation ${\mathbb N}$ for the set of positive integers, Euclidean space ${\mathbb R}^n$ , sphere in ${\mathbb R}^n$ , d-dimensional Lebesgue measure $\mu_d$ , integration over $\mu_d$ .

Background: Let $S^d$ be a sphere in ${\mathbb R}^{d+1}$ normalised such that $\mu_d(S^d) = 1$ . A set of points $x_1, \dots, x_N \in S^d$ is called a spherical t-design if equality $\int_{S^d} P(x) d \mu_d(x) = \frac{1}{N}\sum\limits_{i=1}^N P(x_i)$ holds for all polynomials P in $d+1$ variables, of total degree at most t.

The Theorem: On 22nd September 2010, Andriy Bondarenko, Danylo Radchenko, and Maryna Viazovska submitted to arxiv a paper in which they proved that for every $d\in {\mathbb N}$ there exist a constant $c_d>0$ such that for each $t\in {\mathbb N}$ and each $N\geq c_d t^d$ , there exists a spherical t-design in $S^d$ consisting of N points.

Short context: The concept of a spherical design was introduced by Delsarte, Goethals, and Seidel in 1977, and, as follows from the definition, it is useful for evaluating integrals. Of course, the smaller N, the easier to compute $\frac{1}{N}\sum\limits_{i=1}^N P(x_i)$ . In 1993, Korevaar and Meyers conjectured the existence of spherical t-designs in $S^d$ with as little as $c_d t^d$ points, which is optimal up to the constant factor. The Theorem confirms this conjecture.

Links: Free arxiv version of the original paper is here, journal version is here.

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There are (C_k+o(1))N^2/log(N)^k k-term arithmetic progressions of primes at most N

You need to know: Prime numbers, arithmetic progression, logarithm, limit, infinite product.

Background: We denote by $o(1)$ any function $f(N)$ such that $\lim\limits_{N\to\infty} f(N)=0$ . Let ${\cal P}$ denote the set of all prime numbers.

The Theorem: On 21st September 2010, Ben Green, Terence Tao, and Tamar Ziegler submitted to arxiv a paper which, among other results, finishes the proof of the following theorem. For any fixed integer $k\geq 2$ , the number of k-tuples of primes $p_1<p_2<\dots<p_k \leq N$ which lie in arithmetic progression is $(C_k+o(1))\frac{N^2}{\log^k N}$ , with $C_k=\frac{1}{2(k-1)}\prod\limits_{p\in{\cal P}} \beta_p$ , where $\beta_p=\frac{1}{p}\left(\frac{p}{p-1}\right)^{k-1}$ if $p\leq k$ and $\beta_p=\left(1-\frac{k-1}{p}\right)\left(\frac{p}{p-1}\right)^{k-1}$ if $p\geq k$ .

Short context: In a paper submitted in 2004, Green an Tao proved that, for any k, the set of primes contains infinitely many arithmetic progressions of length k. The Theorem counts how many such progressions exist. In a paper submitted in 2006, Green and Tao proved it for $k=4$ and also proved that the general case (and in fact much more general theorem called “The generalised Hardy-Littlewood conjecture in finite complexity case”) would follow from two new conjectures they formulated. They then, together with Ziegler, proved these conjectures, which finishes the proof of the Theorem.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.8 of this book for an accessible description of the Theorem.

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The absolute values of the coefficients of the chromatic polynomial form a log-concave sequence

You need to know: Graph, proper colouring of vertices of a graph, polynomial.

Background: A sequence $x_0, x_1, \dots, x_n$ of real numbers is called log-concave if $x_{i-1}x_{i+1} \leq x_i^2$ for all $0<i<n$ . For a finite graph G and integer $q>0$ , let $\chi_G(q)$ be the number of proper colouring of vertices of G with q colours. It is known that $\chi_G(q)$ is in fact a polynomial $\chi_G(q)=\sum\limits_{k=0}^n (-1)^{n-k}a_k q^k$ , where $a_0, a_1 \dots, a_n$ are non-negative integers.

The Theorem: On 27th August 2010, June Huh submitted to arxiv a paper in which he proved that if $\chi_G(q)=\sum\limits_{k=0}^n (-1)^{n-k}a_k q^k$ is the chromatic polynomial of a graph G, then the sequence $a_0, a_1 \dots, a_n$ is log-concave.

Short context: The chromatic polynomial was introduced by Birkhoff in 1912, and is one of the most beautiful objects of study in graph theory. In 1968, Read conjectured that the sequence of the absolute values of the coefficients of the chromatic polynomial is unimodal (that is, $a_0 \leq a_1 \leq \dots \leq a_{i-1} \leq a_i \geq a_{i+1} \geq \dots \geq a_n$ for some $0 \leq i \leq n$ ). In 1971, Rota conjectured that this sequence is in fact log-concave and observed that this would imply the Read’s conjecture. The Theorem confirms this prediction. In fact, Rota conjectured an even more general statement, which was also proved in a later work.

Links: Free arxiv version of the original paper is here, journal version is here.

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For integers a>b>0, the the greatest prime factor of a^n-b^n grows superlinearly in n

You need to know: Prime numbers, logarithms, limits. In addition, you need the definition of Lehmer sequence (see here) to understand the last sentences in the context.

Background: For any integer $m$ let $P(m)$ denote the greatest prime factor of $m$ , with the convention that $P(m)=1$ when $m$ is $1$ , $0$ , or $-1$ .

The Theorem: On 6th August 2010, Cameron Stewart submitted to arxiv a paper in which he proved that for any integers $a>b>0$ there is a constant $N=N(a,b)$ , such that $P(a^n-b^n) > n\exp\left(\frac{\log n}{104 \log\log n}\right)$ for all $n\geq N$ .

Short context: In 1886, Bang proved that $P(a^n-1)\geq n+1$ for any integers $a>1$ and $n>2$ . Can this lower bound be improved? In 1965, Erdős conjectured that $\frac{P(2^n-1)}{n}\to\infty$ as $n\to\infty$ . The Theorem (with $b=1$ ) proves Erdős conjecture (for any $a>1$ in place of $2$ ) and gives the first improvement on Bang’s bound for 124 years. For general $a>b>0$ , Zsigmondy proved in 1892 that $P(a^n-b^n)\geq n+1$ for $n>2$ , and Schinzel asked in 1962 if $P(a^n-b^n)\geq 2n$ for large n. The Theorem provides a much better lower bound than $2n$ . In fact, Stewart proved, much more generally, that the same lower bound holds for $P(u_n)$ , where $u_n$ is the n-th term of Lehmer sequence. Previously, it was only known (as a corollary from this theorem) that $P(u_n) \geq n-1$ for $n>30$ .

Links: Free arxiv version of the original paper is here, journal version is here.

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Algorithmic version of general Lovász local lemma is true

You need to know: Probability, probability space, event, random variable, mutually independent random variables.

Background: Let ${\cal P}$ be a finite collection of mutually independent random variables in a fixed probability space $\Omega$ . Let $S$ be a finite family of events in $\Omega$ determined by ${\cal P}$ . For $A\in S$ , let $v(A)$ be the (unique) minimal subset of ${\cal P}$ that determines $A$ . For $A\in S$ , let $\Gamma_S(A)$ be the set of $B\in S$ such that $A\neq B$ but $v(A)\cap v(B) \neq \emptyset$ . We say that an evaluation of the variables in ${\cal P}$ violates $A\in S$ if it makes A happen.

The Theorem: On 3rd March 2009, Robin Moser and Gábor Tardos submitted to arxiv a paper in which they proved that if there exists an assignment of reals $x:S \to (0,1)$ such that $\forall A\in S: \text{Prob}[A]\leq x(A) \prod\limits_{B\in \Gamma_S(A)} (1-x(B))$ , then there exists an assignment of values to the variables in ${\cal P}$ not violating any of the events in S. Moreover, this assignment can be found by a randomised algorithm with the expected number of resampling steps at most $\sum\limits_{A\in S}\frac{x(A)}{1-x(A)}$ .

Short context: The Theorem without “moreover” part is (a slightly modified version of) famous Lovász local lemma, discovered by Erdős and Lovász in 1975, which allows one to show the existence of certain objects occurring with exponentially small probability. The “moreover” part of the Theorem can be used to actually construct such objects. Previously, algorithmic versions where developed for some applications of the Lovász local lemma, but not for the lemma in general.

Links: Free arxiv version of the original paper is here, journal version is here.

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The least squares mean for positive definite matrices is monotone

You need to know: Matrix, square matrix, symmetric matrix, eigenvalue of a square matrix, matrix multiplication, inverse $A^{-1}$ of matrix A.

Background: Let $\lambda_i(A), i=1,\dots,n$ denote the eigenvalues of $n \times n$ symmetric matrix A. If all $\lambda_i(A)$ are non-negative, A is called positive semidefinite, while if all $\lambda_i(A)$ are positive, A is called positive definite. Let $P_n$ be the set of all positive definite $n \times n$ matrices. For $A,B \in P_n$ , we write $A\geq B$ if $A-B$ is positive semidefinite. The trace metric distance between $A,B \in P_n$ is $\delta(A,B)=\sqrt{\sum\limits_{i=1}^n \log^2 \lambda_i(A^{-1}B)}$ . The least squares mean (also known as Karcher mean) of k matrices $A_1, \dots, A_k \in P_n$ is $\sigma(A_1, \dots, A_k)=\arg \min\limits_{X\in P_n}\sum\limits_{i=1}^k \delta^2(X,A_i)$ .

The Theorem: On 27th June 2010, Jimmie Lawson and Yongdo Lim submitted to arxiv and Mathematische Annalen a paper in which they proved that if $B_i \leq A_i$ , $i=1,\dots,k$ , then $\sigma(B_1, \dots, B_k) \leq \sigma(A_1, \dots, A_k)$ .

Short context: The concept of least squares mean is a natural generalisation of the geometric mean that has been developed by Grove and Karcher in 1973 for general metric spaces, and applied to positive definite matrices by Moakher in 2005. The property proved in the Theorem is called monotonicity of the least squares mean, and was conjectured by Bhatia and Holbrook in 2006.

Links: Free arxiv version of the original paper is here, journal version is here.

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